So this looks like an awkward design fault. They overdid in the strive for low gate voltage 
I just wonder why the comparator outputs those 3.75 volts rather than something closer to 5 Volt...
edit
I see.. the LM358 is not a true rail-to-rail opamp. Is goes down to ground but not to Vcc.. Max input is Vcc -1,5 volts = around 3.5volts, which is than buffered by the voltage follower..... This works in combination with a "typical" logic level MOSFET but not with that super low gate voltage type.. As they say: "The better is the enemy of good "
So another workaround would be to change to a pin compatible rail to rail dual opamp....